Weighted ergodic theorems for mean ergodic Lj-contractions

It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any LI-contraction with mean ergodic (ME) modulus, and for any positive contraction of L_p with 1 < p < ∞. We extend the return times theorem by proving that if 5 is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any g bounded measurable {Sⁿg(ω)} is a universally good weight for a.e. ui. We prove that if a bounded sequence has "Fourier coefikents", then its weighted averages for any Lj-contraction with mean ergodic modulus converge in Li-norm. In order to produce weights, good for weighted ergodic theorems for Li-contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tensor product of Li-contractions is the product of their moduli, and that the tensor product of positive quasi-ME L₁-contractions is quasi-ME.